I came across the following two definitions in a book about Integer Programming:
Definition 1.1 A subset of $R^n$ described by a finite set of linear constraints $P=\{x \in R^n: Ax \leq b \}$ is a polyhedron.
Definition 1.2 A polyhedron $P \subseteq R^{n+p}$ is a formulation for a set $X \subseteq Z^n \times R^p$ if and only if $X=P \cap (Z^n \times R^p)$.
I don't really understand definition 1.2. Could anyone explain to me in simple language what this tries to say? I especially don't get this: '$X=P \cap (Z^n \times R^p)$'; what does it mean to take the intersection of a set and a Cartesian product (I know what a Cartesian product is but how can that intersect with a regular set)?
When unpacking definitions, it often helps to think of a very simple example. Let's take the case $p=0$ and $n=2$. Think of $\mathbb{Z}^2$ as the lattice points embedded in the Euclidean plane $\mathbb{R}^2$. Now take $X \subset \mathbb{Z}^2$ be a very specific set, say the corners of a square, $X=\{(0,0),(1,0),(1,1),(0,1)\}$.
Now graph the region $P$ satisfying the inequalities $$ 0 \leq x \leq 1 \\ 0 \leq y \leq 1. $$ Then $P$ is a square in the plane $\mathbb{R}^2$ and $X = P \cap \mathbb{Z}^2$. So the region $P$ contains precisely the lattice points in the set $X$. We call $P$ or the system of inequalities a formulation for the set $X$. But there are many formulations, for instance the region $P'$ given by $$ 0 \leq x \leq 1.1 \\ 0 \leq y \leq 1.9. $$ Now come up with examples where $p>0$.